How can I make lens and prism simulations in Blender Game Engine for optics experiments?
Game engines are vastly different to rendering engines. Rendering engines are optimized for accuracy, game engines for speed.
Caustics (refraction) is something that is incredibly computationally expensive. It's only in the last decade that it is possible to do it on a home computer, and even then it is slow.

Do my knowledge, not a single commercially or freely available game engine supports real-time caustics accurate enough to be used as a demonstration.
Rendering out preset movies and using buttons to switch between them can be done really easily....

That said, with some creative coding, there is no reason it can't be faked really well, but you'd have to be really heavily into your math for that.